The geomorphic structure of the runoff peak

Hydrol. Earth Syst. Sci., 15, 1853–1863, 2011www.hydrol-earth-syst-sci.net/15/1853/2011/doi:10.5194/hess-15-1853-2011© Author(s) 2011. CC Attribution 3.0 License.

Hydrology andEarth System

Sciences

The geomorphic structure of the runoff peak

R. Rigon1, P. D’Odorico2, and G. Bertoldi3

1Dipartimento di Ingegneria Civile e Ambientale and CUDAM, Universita di Trento, Via Mesiano 77, Trento, Italy2Department of Environmental Sciences, University of Virginia, 291 McCormick Road, P.O. Box 400123, Charlottesville,VA 22904-4123, USA3Institute for Alpine Environment, EURAC Research, Viale Druso 1, Bolzano, Italy

Received: 23 December 2010 – Published in Hydrol. Earth Syst. Sci. Discuss.: 24 January 2011Revised: 10 May 2011 – Accepted: 25 May 2011 – Published: 17 June 2011

Abstract. This paper develops a theoretical framework toinvestigate the core dependence of peak flows on the geo-morphic properties of river basins. Based on the theory oftransport by travel times, and simple hydrodynamic charac-terization of floods, this new framework invokes the linearityand invariance of the hydrologic response to provide analyt-ical and semi-analytical expressions for peak flow, time topeak, and area contributing to the peak runoff. These re-sults are obtained for the case of constant-intensity hyeto-graph using the Intensity-Duration-Frequency (IDF) curvesto estimate extreme flow values as a function of the rain-fall return period. Results show that, with constant-intensityhyetographs, the time-to-peak is greater than rainfall durationand usually shorter than the basin concentration time. More-over, the critical storm duration is shown to be independentof rainfall return period as well as the area contributing tothe flow peak. The same results are found when the effectsof hydrodynamic dispersion are accounted for. Further, it isshown that, when the effects of hydrodynamic dispersion arenegligible, the basin area contributing to the peak dischargedoes not depend on the channel velocity, but is a geomorphicpropriety of the basin. As an example this framework is ap-plied to three watersheds. In particular, the runoff peak, thecritical rainfall durations and the time to peak are calculatedfor all links within a network to assess how they increasewith basin area.

Correspondence to:R. Rigon([email protected])

1 Introduction

A number of hydrological analyses require the evaluation ofthe highest peak-flow values expected to occur with a givenreturn period. Most of the methods addressing this issue –from the simple rational methodMulvaney(1851), Doodge(1957) to the use of distributed rainfall-runoff models (e.g.,Beven, 2001) – have been developed with the purpose of pro-viding quantitative predictions of peak flows for engineeringapplications more than a synthesis of their dependence onthe geomorphic and hydrodynamic characteristics of the wa-tershed. To this end, this paper develops a simplified the-ory based on the concepts of geomorphologic instantaneousunit hydrograph (GIUH) and of width function (Rinaldo etal., 1991, 1995; D’Odorico and Rigon, 2003). This theoryextends the results ofHenderson(1963) and Meynink andCordery(1976) and complements some findings byRobin-son and Sivapalan(1997).

In this paper we analyze the critical rainfall duration forlinear systems as, for instance, inFiorentino et al.(1987)andIacobellis and Fiorentino(2000) who found that the flowpeak has a linear dependence on the rainfall excess intensityover a duration equalling the IUH lag-time. Our study pro-vides a geomorphic interpretation of these previous results.To this end, we use the framework of the GIUH theory todetermine the rainfall duration that causes the highest peak -flow.

The goals of the paper include: understanding the geo-morphic structure of the highest peak-flow caused by rain-fall with given return period; redefining the concept of con-centration time within the framework of the GIUH theory;determining the duration of the rainfall able to generate themaximum peak flow under the assigned climatic conditions;determining the time to peak as a function of rainfall andbasin characteristics.

Published by Copernicus Publications on behalf of the European Geosciences Union.

http://creativecommons.org/licenses/by/3.0/

1854 R. Rigon et al.: The geomorphic structure of the runoff peak

1.1 Basic concepts and results

The concept of Unit Hydrograph (Sherman, 1932) is usedin the representation of the hydrograph as a sum of the re-sponses to different rainfall inputs observed throughout anindividual rainstorm

Q(t) = AT

∫ t

0f (t − τ) p(τ) dτ (1)

with AT being the total contributing area,t the time,τ thetime counted starting at the beginning of the rainstorm,p theintensity of effective precipitation at timeτ , andf (t) theinstantaneous unit hydrograph (IUH). The IUH representsthe travel time probability density function of a unit amountof water instantaneously injected into the basin (Gupta andMesa, 1988); f (t) can be determined either through somesimple conceptual frameworks e.g. (Nash, 1957) or throughthe geomorphological theory (Rodriguez-Iturbe and Valdes,1979; Gupta et al., 1980). Depending on the physical hy-potheses underlying the different formulations of the IUH,f (t) can be defined either within an infinite or a finite timedomain. In the latter casef (t) = 0 for t ≥ τc, whereτc isthe concentration time of the rational method (i.e. the timeat which the whole basin contributes to the discharge at theoutlet).

The integral:

S(t) =

∫ t

0f (τ) dτ (2)

is known in literature asS-hydrograph (e.g.,Doodge, 2003,p. 86) and represents the cumulative probability distributionof travel times inside the basin (S(t) = 1 for t ≥ τc), andS(t) can be interpreted as the ratio between contributing areaat timet and basin area, sinceAT in Eq. (1) is the total con-tributing area.

In this simplified approach the rate of (effective) precipi-tation is assumed to be constant throughout individual rain-storms of durationtp:

p(t, tp) ≡ p H(tp − t) H(t), (3)

whereH() is the Heaviside step function (i.e.H(t) = 1 fort ≥ 0 andH = 0 otherwise). By definition,p can be con-sidered the expected value of the effective intensity of rain-fall during the storm. In what follows we will refer to theexpected value ofp as a first-order approximation of the“real” storm hyetograph. This approach will allow us to ob-tain semi-analytical results that could be easily generalized tothe case of hyetographs with non-constant intensity as sug-gested byD’Odorico et al.(2005) for the case of landslide-triggering precipitation.

The time-to-peak,t∗, can be found by solving the follow-ing equation, first derived byHenderson(1963) and hereaftercalledHenderson’s equation(see also Appendix A):

f (t) = f(t − tp

)tp ≤ τc. (4)

The graphic solution of Eq. (4) is illustrated in Fig. 1. Fort > τc, S(t) is a constant (hence,f (t) = dS(t)/dt = 0), whileS(t − tp) is an increasing function oft (i.e. f (t − tp) > 0).Therefore the solution,t∗, of Eq. (4) needs to be searched inthe interval[tp, τc]. Figure 1, shows the existence of a de-lay, (1t = t∗ − tp), between the end of the rainstorm and theoccurrence of the flow peak. This delay depends on the char-acteristics of the IUH and its parameters. The delay,1t , cor-responding to the main (i.e. highest) peak can be easily deter-mined: for example, in the case of the catchment discussedin Sect. 3,1t is a decreasing function oftp and becomes nullas tp approachesτc (Fig. 2). Moreover, Fig. 2 shows thatfor small values oftp multiple peak flows – correspondingto secondary maxima – may occur.Henderson(1963) andRobinson and Sivapalan(1997) solved Eq. (4) using a tri-angular hydrograph. However it can be solved analytically(Appendix B) in the case of Nash hydrograph (Nash, 1957)and with simple numerical code in the general case, as ex-plained in the following section.

If t∗ is the time to peak – counted from the beginning of therainfall – the peak flow,Qp, is then estimated asQ(t∗) usingEq. (A5) for the case of hyetographs with constant intensity:

Qp(t∗) =

{p AT

(S(t∗) − S

(t∗ − tp

))= p A∗ 0 ≤ tp ≤ τc

p AT S(τc) = p AT tp > τc(5)

with t∗ being a function oftp (through Eq.4) and S(t∗)

the fraction of contributing area att = t∗. If the durationis smaller than the concentration time, the contributing areaat t = t∗ is A∗

= AT [S(t∗)−S(t∗ − tp)].

1.2 Extreme values of peak flows

The maximum peak flow occurring after a rainstorm witha certain return period,tr, (hereafter referred to as extremepeak flow) can be determined by expressingp in Eq. (5) asa function of duration,tp, and return period,tr. These curvesprovide a statistical representation of the most severe rainfallconditions for a certain geographic location and return pe-riod. Thus, the extreme peak-flow discharge,Qp (Eq. 5),depends ontp also throughp. Because for any given re-turn period,p = p(tp|tr) is a decreasing function oftp andS(t∗) = S(tp+1t) is an increasing function of its argument,there is a particular rainfall duration,t∗p , which maximizesthe peak-flow discharge. Such a duration needs to be shorterthan (or equal to) the concentration timeτc. This criticalduration can be found by solving the equation dQp/dtp =

dQ(tp +1t)/dtp = 0, where1t = 1t(tp) is a smooth func-tion of tp. We will indicate with1t∗ the value of1t(tp)

calculated fortp = t∗p . The first order-derivative of Eq. (5a)becomesd Q(t∗)

dtp= AT

{p′

(t∗p |tr

) [S(t∗p + 1t∗

)− S

(1t∗

)]+ p

(t∗p |tr

) [f(t∗p + 1t∗

)(1 + 1′(t))

− f(1t∗

)1′(t)

] }= 0 (6)

Hydrol. Earth Syst. Sci., 15, 1853–1863, 2011 www.hydrol-earth-syst-sci.net/15/1853/2011/

R. Rigon et al.: The geomorphic structure of the runoff peak 1855

Fig. 1. The solutions of Eq. (4) are given by the crossing of the unithydrographf (t) with another unit hydrograph,f (t −tp), shifted bya distance,tp. The figure shows:(a) the graphical solution of theequation (in blackf (t) and in greyf (t−tp)); t∗ is the time to peak.(b) t∗ is usually larger thantp. This is true for constant intensityuniform hyetograph;(c) the discharge obtained by the convolutionof the IUH in (a) with the rainfall in(b).

wherep′(t∗p |tr) is the first-order derivative of thep(t∗p |tr) withrespect totp and the first-order derivative ofS(t) has beenexpressed as the IUH (i.e.S′(tp+1t) = f (tp+1t)). Substi-tuting Eq. (4) into Eq. (6) we obtain

p′(tp|tr

)p(tp|tr

) = −f(tp + 1t

)S(tp + 1t

)− S(1t)

. (7)

which is equivalent to the main equation inMeynink andCordery(1976), though it is here derived for any shape ofthe IUH. We use intensity-duration-frequency (IDF) curvesto relate rainfall intensity. Scaling models of IDF curves areoften based on power laws

p(tp|tr

)= a(tr) t−m

p (8)

where a(tr) is a function of the return periodtr and0 ≤ m ≤ 1 is independent oftr. Equation (8) in Eq. (7)gives

m =tp f (t∗)

S(t∗) − S(1t∗)≡ g

(t∗p

)(9)

wheret∗ = t∗p +1t∗. Because the dependence ontr in Eq. (8)is througha(tr) – which does not appear in Eq. (9) – Eq. (9)implies that, according to this linear theory of the hydrologicresponse, the critical rainfall duration (t∗p ) associated with ex-treme runoff peak flow values, is independent of the returnperiod. The same result applies also to the case of self- simi-lar design storm hyetograph (e.g.,Burlando and Rosso, 1996)and to the design storm hyetograph suggested inHershfield(1961) and commonly used in the engineering practice in theUS sinceBell (1969).

The solution of the set of Eqs. (9) and (4) can be usedto determine the critical rainfall duration,t∗p , and the timeto peak,t∗. It can be shown thatg(0) = 1 andg(∞) = 0,consistently with the common observation that values ofm

span the interval[0,1] (m ranges most commonly between0.5 and 1). Depending on the rainfall durationtp and onthe shape of the IUH, for some values ofm, it is possible tohave multiple solutions of Eq. (9), which correspond to localminima or maxima of discharge, as shown on a case study inSection3

Oncet∗p is known from Eq. (9), Eq. (5) provides the maxi-mum discharge under a rainfall of assigned return period,tr,as

QMAX (tr) = p(t∗p |tr

)C(t∗, t∗p

)AT (10)

where C(t∗, t∗p ) ≡ S(t∗) − S(t∗ − t∗p ). We notice thatEq. (10) is similar to the well known rational method equa-tion Chow et al.(1988); however, the runoff coefficient,C,depends on the effective fraction of contributing area eval-uated as a function of time to peak,t∗, and on the criticalrainfall duration,t∗p (Eq. 6), rather than on the concentra-tion time. The total contributing area,AT, can be deter-mined as explained in the following section. We note that inthe rational method the coefficientC accounts for effects of”within-storm” rainfall variability, runoff-generation (i.e.,Cis a runoff coefficient), and runoff routing. Our approach ex-plains only the dependence ofC on routing processes, whilethe effects of “within-storm” rainfall patterns are not inves-tigated. We also note that, unlike the rational method, ourtheory does not assume values of contributing area and rain-fall duration. Rather, bothAT andt∗p are the outcome of the

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Fig. 2. Delay,1t , of time to peak with respect to the end of the rainstorm, as a function of rainfall duration,tp, for the case study of Longowatershed (Italy), illustrated in detail Paragraph3. 1t is a decreasing function oftp and for valuestp ≥ τc, the delay is null. For smallertpmultiple peak flows are possible. The grey points correspond to secondary peak flows and the black points to the largest peak flow.

interplay of basin and climatic characteristics, and are calcu-lated as values associated with the maximum peak flows.

2 The geomorphological analysis of runoff peaks

The geomorphological theory of the hydrologic response(Rodriguez-Iturbe and Valdes, 1979; Gupta et al., 1980; Ri-naldo et al., 1991; Rinaldo and and Rodriguez-Iturbe, 1996)provides an interpretation of the IUH, based on the basinmorphology and simple dynamical assumptions. In this pa-per we express the Geomorphologic Instantaneous Unit Hy-drograph (GIUH) through a generalization of the width func-tion, W(x), (e.g.,Shreve, 1969; Kirkby, 1986; Gupta andMesa, 1988; Brutsaert, 2005). W(x) is the probability distri-bution of distances,x (measured along the network), betweenany point in the basin and the outlet. In recent years a numberof studies have recognized the soundness of this approach,strengthened its theoretical bases (Rinaldo et al., 1991, 1995;Saco and Kumar, 2002a,b; D’Odorico and Rigon, 2003; Bot-ter and Rinaldo, 2003) and shown its applicability and cal-ibration to small and large catchments (e.g.,Naden, 1992;Snell and Sivapalan, 1994; Franchini and O’Connell, 1996;Da Ross and Borga, 1997; Naden et al., 1999; Yang et al.,2002; Brutsaert, 2005).

The basin-scale travel time distribution,f (t), can be ex-pressed as a function ofW(x)

f (t) =

∫ L

0W(x) f (t |x) dx (11)

wheref (t |x) is the travel time distribution in a path of lengthx, andL is the length of the longest drainage path.

2.1 The kinematic case

When the effects of hydrodynamic dispersion are negligible,water is subject mainly to advection (kinematic wave), andthe probability distribution of travel times,f (t |x), for therain falling at a distancex (measured along the flow path)from the outlet is

f (t |x) = u δ(t −

x

u

)(12)

with u being the flood wave celerity andδ() the Dirac delta-function. Hence, Eq. (12) in Eq. (11) gives:

f (t) = u W(u t). (13)

In this framework, which generalizesRoss(1921), the con-centration time is rigourouslyτc = L/u, with L being thelongest drainage path. When the IDF curves are expressedby Eq. (8) andf (t) by Eqs. (13), (4) and (9) become:

W(u t) = W(u(t − tp

)). (14)

m =u tp W

(u(tp + 1t

))(S(tp + 1t

)− S(1t)

) (15)

Equation (14) provides the lag1t = t∗ − tp between the endof the storm and the peakflow occurrence, while Eq. (15)gives the critical rainfall duration,t∗p . As noted before,t∗pis independent oftr. When Eqs. (14) and (15) are solved for



different values of the parameteru, the drainage area contri-bution to the peak flow,A∗

≡ AT(S(t∗)−S(t∗ − tp)), doesnot change. As shown in Figure 3,f (t) shrinks in width asu increases, but at the same time it increases in height, main-taining a constant area,A∗

≡ AT(S(t∗)−S(t∗ − tp)).The maximum peak discharge depends hyperbolically on

rainfall durations (and flow velocity):

QMAX = a(tr)(t∗p

)−m

C(t∗, t∗p

)AT = a(tr)

(t∗p

)−m

A∗. (16)

Equation (13) provides a model of IUH which depends onlyon the parameteru and on the basin morphology (i.e. on theshape of the width function). BecauseA∗ is independentof u, bothA∗ andQMAX depend only on the form ofW(),i.e. on the structure of the flow paths.

2.2 The effect of diffusive wave propagation

The analysis presented in the previous section accounts forthe mean and most of the variance of the unit hydrograph(e.g.,D’Odorico and Rigon, 2003). However, it does not in-clude the effects of hydrodynamic dispersion which intro-duces a smoothing on the peak flows. In this case floodrouting can be expressed by a parabolic differential equationwhich is obtained either as a diffusive-wave approximation ofthe de Saint-Venant equations, or through the assumption thatwater parcels are subject to Wiener dynamics, superimposedto the deterministic advection discussed in the previous sec-tions. This assumption leads to the estimation of the traveltime distribution as a solution of the Kolmogorov’s backwardEq. (Mesa and Mifflin, 1986) with suitable boundary con-ditions. Thus, the probability distribution of travel times isexpressed by the Gaussian inverse function (Rinaldo et al.,1991), defined now fort ∈ [0, ∞]

f (t |x) =x

√4 π D t3

exp

[−

(x − u t)2

4 D t

](17)

whereD is the coefficient of hydrodynamic dispersion; thekinematic case (Eq.12) is obtained forD → 0.

In this case the hydrograph can be expressed (see Eq.C2in Appendix C) as

Q(t)

AT p(tp, tr

) ={∫ L

0 dx W(x) 2(t |x) if 0 ≤ t ≤ tp∫ L

0 dx W(x)[2(t |x) − 2

(t − tp|x

)]if t > tp,

(18)

where

2(t |x) =1

2Erfc

(x

2√

t D−

u

2

√t

D

)+

1

2Exp

(u x

D

)Erfc

(x

2√

tD+

u

2

√t

D

). (19)

Notice that d2(t |x)/dt = f (t |x) with f (t |x) given byEq. (17); therefore, fort ≥ tp, Henderson’s equation can bewritten as∫ L

0dx W(x)

[f (t |x) − f

(t − tp|x

)]= 0. (20)

The timet∗ satisfying Eq. (20) is always larger than theprecipitation duration as in the case discussed in Sect. 2.The other results developed in the previous section can beextended to the caseD > 0, once the functionW(u t) inEqs. (13), (14) and Eq. (15) is replaced by

ω(t) ≡

∫ L

0W(x) f (t |x) dx. (21)

Thus, the critical rainfall timet∗p can be calculated by in-serting Eq. (21) into Eqs. (14) and (15). It is found thatthe area,A∗, contributing to the flow peak is independentof the return period. However, whenD assumes large values(∼ 1000m2 s−1), it can be shown thatA∗ depends onD, theshape ofW(x), and onu.

The peak discharge can be obtained numerically by sub-stituting the time to peak,t∗, into Eq. (18). Interestingly,the concept of concentration time, used in the non-diffusive(kinematic) framework, would be meaningless in this casebecause the domain ofω(t) andf (t |x) are infinite. How-ever, it can be here re-introduced as a stochastic variable,τc with distribution given by Eq. (17). In fact, in the caseD 6=0, after a timeτc = L/u has elapsed from the beginningof the rainstorm, the furthermost portions of the basin maystill contribute to the hydrologic response with probabilitysmaller than 1.

3 Case study

An application of the theory developed in this paper, wascarried out for the cases of the Longo watershed, a smallalpine catchment (A = 10.3 km2) within the Avisio basin(A = 469 km2) at Predazzo (Italy) and of the Salt River inCalifornia (USA) (A = 2020 km2).

In this application the width function was calculated differ-entiating for the velocities in channels,uc, and in hillslopes,uh, (Rinaldo et al., 1995; D’Odorico and Rigon, 2003), andintroducing a rescaled-width function,W ′(x′), which is de-fined by measuring the length of the drainage paths as sumof the portion inside the river network,xc, and that acrossthe hillslope,xh, with the latter being amplified by a factorr = uc/uh

x′= xc +r xh. (22)

The geomorphologic unit hydrograph can be thus expressedas

f (t) =1

√4πDt3

∫ L

0x′W ′(x′) e−(x′

−uct)2/4Dtdx′ (23)



2000 4000 6000 8000 10000t !s"5

101520253035

u W#u!t$ ! 10"5u#2 m%s

u#1 m%s

Fig. 3. Two examples unit hydrographs derived from the width function withu = 2 m s−1 (with higher peak on the left) andu = 1 m s−1

(with smaller peaks on the right). The area contributing to the maximum discharge, as derived from the Henderson’s equation and Eq. (17),is 80 % of the total contributing area for each of the hydrograph and marked in grey (darker for the caseu = 1 and of an intermediate tonefor both the cases). In theu = 1 m s−1 case, the rainfall duration which gives(Q1)MAX = 14 m3 s−1 is (t1)∗p = 5828 s; is also(t1)∗ = 6808 s

(the right limit of the grey). In the caseu = 2 m s−1 it is: (Q2)MAX = 21.2 m3 s−1, (t2)∗p = 2914 s,(t2)∗ = 3404 s.

using the rescaled width function to express the probabilitydistribution of drainage paths as if it was in a fictitious net-work where flood waves propagate at constant velocity,uc.

Furthermore, different moisture conditions (D’Odoricoand Rigon, 2003) are expressed through the quantiles,q, of awetness index distribution (Beven and Kirkby, 1979; Barlinget al., 1994), which defines the parts of the basins contribut-ing to the hydrograph as saturation overland flow (e.g.,Siva-palan et al., 1987; Beven and Wood, 1993; Woods and Siva-palan, 1997, 1999). Thus, the total contributing area,AT,is a function of the degree of saturation of the watershed,AT = q Ab, whereAb is the basin area; the width functionis calculated using only the saturated part of the basins (andnot the whole basin area) and depends on the channel veloc-ity, uc, the scale parameterr and the saturated fraction,q, ofthe basin, while the area contributing to the peak flow,A∗,depends also onq.

Figure 4 provides the graphical solution (i.e.,t∗p ) of Eq. (9),for the case of the Longo basin, whereD = 0, and for thevalue of the IDF curvem = 0.63, found by analyzing theextreme precipitation in the area. It illustrates a typical de-pendence ofg(tp) on tp (with 1t(t∗) given by Eq.4). Thenon-monotonic decrease makes clear the possibility to havemultiple solutions of Eq. (9), which correspond to local min-ima or maxima of discharge. The Figure shows also howby increasing the velocity the time to peak decreases and thepeak discharge increases. However the area contributing tothe peak flow remains constant.

Table 1 reports the values of the variablesτc, t∗, t∗p , A∗

andQ∗ in the Longo basin for different values of the satu-rated fraction of the basin. It is observed thatτc, non-linearlyincreases with increasing values ofq. It is also observedthat t∗ is usually less than halfτc. This difference is dueto the long tails of the width function (D’Odorico and Rigon,2003). The area contributing to the peak is about eighty percent of the total saturated area,AT, and this fraction remainsalmost constant with varyingq, i.e. a first rough estimationgives:

A∗∼ 0.8 AT ∼ 0.8 q Ab (24)

Excluding the lowest saturation conditions, the critical rain-fall duration increases almost linearly withq > 20 %, whilethe delay1t∗ of the maximum peak increases non-linearly.

Because of its simplicity, the flow peak analysis can beeasily extended to estimate the runoff peak in all the chan-nel network links inside a basin, providing a regionalizationof peak flows. Figure 5a, b and c show an example for theAvisio basin. Figure 5a shows how the maximum dischargeQMAX (with rainfall return period,tr = 100 yr) at any linkincreases with the contributing areaA, as:

QMAX (A) ≈ 0.906A0.984 (25)

where the discharge is in m3 s−1 and the contributing areain km2. In this case, the parameters (q = 30 %, uc =

2 m s−1, r = 100) were derived from calibration on a few



Fig. 4. Graphical solution of Eq. (9) for three values of channels celerityuc. The value ofm = 0.63 (horizontal line) is found by analysing theextreme precipitation in the area of the Longo watershed (Italy); the other curves represent the functiong(tp) given by Eq. (9) with differentchannel velocities. By increasing the velocity the time to peak decreases and the peak discharge increases. However the area contributing tothe peak flow remains constant in all of the three cases.

Table 1. Relevant quantities calculated for the rescaled width func-tions of the Longo catchment (r = 10):q is the fraction of saturatedareas;t∗p is the duration of the rainfall which gives the largest peakdischarge; t∗ the time to peak;τc the concentration time;A∗ thearea contributing to the peak;Q∗ is the largest discharge;AT thetotal saturated area.

t∗p t∗ τc A∗ Q∗ AT

q (s) (s) (s) (km2) (m3 s−1) (km2)

0.05 4624 5075 14 631 0.418 6.25 0.5290.13 5834 6808 15 721 1.086 14 1.3730.28 6063 7632 16 287 2.288 28.85 2.9310.55 6407 8325 16 570 4.497 54.8 5.6680.82 6763 8884 17 496 6.799 80.07 8.472

1 7292 9925 17 496 8.3171 94.012 10.33

high-flow events (with approximatively 100 years of returnperiod) measured for the largest sub-catchments of the Avi-sio. Clearly the discharges obtained are not representativeof small basins (A ≤ 50 km2), in which flow parameters andrainfall should be chosen differently. The heterogeneity ofthe responses shown in Fig. 5 for the small contributing ar-eas is completely due to the different pathway lengths andtheir subdivision between hillslopes and channels. The max-imum discharge at any link is due to rainfall of different du-ration as shown in Figure 5b. The critical duration is only

weakly dependent on (i.e., increasing with) the area. Forsmaller contributing areas (indicated in the Figure as greypoints) the critical rainfall duration is even more affected bythe variability of the hillslope length. Even though data werenot available to confirm the variability oft∗p in the smallerbasins, these results are consistent with those ofWood et al.(1990). Figure 5c shows the delay,1t , of the time to peakwith respect to rainfall duration as a function of the contribut-ing areas. It is found that1t increases non-linearly with thecontributing area. The flow peak is delayed with respect tothe end of the rainstorm and this delay is larger in the largersubbasins.

The effect of the hydrodynamic dispersion is presented inFig. 6a, which showst∗ as a function oftp in a mid-size basin(Salt River – CA, 2020 km2). Notice howt∗ is always largerthantp whentp < τc (in this basinτc=L/u=11.2 h), as opposedto the classic assumptions of the rational method thatt∗ = tpwhen tp < τc. Because from Eq. (5) Qp/(p AT) = A∗/A,Fig. 6b shows the portion of the watershed contributing to thebasin response at the peak flow. It is observed that, with lowvalues of the dispersion coefficient, the response is similar tothe kinematic case and the contributing saturated source areais almostAT (i.e. A∗

≈ AT) when tp ≈ τc = 11.2 h. Withrelatively large values ofD (hence of the variance of traveltimes),A∗/AT remains smaller than 1 in a broader intervalof values oftp. The adimensional parameterQp/(p AT)

is called in literature “the peakdeness” of the hydrograph



10 20 50 100 200 500A Hkm2L5

10

50100

QMAXHm3s-1L

100 200 300 400 500A Hkm2L8

10

12

14

16

18tpHhL

100 200 300 400 A Hkm2L

0.05

0.10

0.15∆t�tp

QMAX ~ A0.98

a)

bL

cL

Fig. 5. (a)Plot of the peak dischargeQMAX for all the links withinthe Avisio basin (Italy) as a function of contributing area. The pa-rameters (q = 30%,uc = 2 m s−1, r = u/uh = 100) were obtainedfrom calibration on some real event in a few subcatchments andusing IDF curves for a return periodtr = 100 yr. (b) Critical rain-fall duration, t∗p , of links ends inside the Avisio basin (Italy). Forsmaller contributing areas (indicated in the Figure as grey points)t∗p andQMAX are affected by the variability of the hillslope length.(c) Delay of the time to peak with respect to the rainfall duration asa function of the contributing areas.

(Meynink and Cordery, 1976) and was found to vary in realcases between 0.5 and 1.5. Values larger than 1 must thenbe due to the variability of the rainfall and not to the basingeomorphology.

0 5 10 15tp HhL0

5

10

15

t* HhL

0 5 10 15tp HhL0.0

0.2

0.4

0.6

0.8

1.0

Qp

AT p

D = 1 m2 s-1

D = 104 m2 s-1

D = 1 m2 s-1

D = 104 m2 s-1

Fig. 6. (top) Time to peak (t∗) and (bottom) normalized peakflow values as a function of the rainstorm duration (tp) in the SaltRiver (CA).

4 Conclusions

The paper has developed a method for the evaluation ofextreme peak-flows based on the theory of the instanta-neous unit hydrograph and on the assumption that stormhyetographs have constant rainfall intensity. The system oftwo Eqs. (14) and (15) gives the maximum allowable dis-charge,QMAX , produced by a precipitation event with a cer-tain return period,tr, time to peak,t∗ = tp+1t(tp) and con-stant intensity. When the intensity-duration dependence ofextreme precipitation is expressed by a power law,t∗p and1t , and the area contributing to the peak do not depend onthe return period. Analytical expressions of1t where ob-tained in particular for the linear reservoir and for the Nashmodels (in AppendixB).

These methods were also applied to the geomorphologicalIUH using the framework of the rescaled width function. Itwas found that, in the kinematic case, the area,A∗, contribut-ing to extreme peak flows,QMAX , does not depend on chan-nel celerity but on the saturated fraction of the basin,q, theratio between channel and hillslope velocities,uc/uh. Whendispersion is introduced, the same area depends not only ondispersionD, also on the velocity in the channels. Thus, theextreme peakflow,QMAX , is expressed through a framework



that generalizes and clarifies the traditional expression of therational method.

It was also shown that the peak discharge due to surfacerunoff increases almost linearly with the contributing area.Moreover, both the critical rainfall duration,t∗p , associatedwith maximum peakflow values, and the delay betweent∗pand the time-to-peak are increasing functions of the con-tributing area. The applicability of this framework is partlyshown through a few basins with different sizes and mor-phologies. The semi-analytical character of the simplifiedtheory allows for a fast estimation of the maximum dischargeflowing in any link of the river network.

Appendix A

When t < tp and the rainfall is given by Eq. (3) the rate offlow is:

Q(t) = AT p

∫ t

0f (t − τ) dτ = −AT p

∫ 0

t

f(t1) dt1

= p AT S(t) (A1)

where the change of variable in the integral is straightforwardandA(t) ≡ ATS(t) is the watershed area contributing to theflood discharge at timet as follows from the definition ofwidth function. Fort > tp we have instead:

Q(t) = AT p

∫ tp

0f (t − τ) d τ (A2)

After the change of variablet1 = t −τ we obtain:

Q(t) = −AT p

∫ t−tp

t

f (t1) dt1

= AT p

[∫ t

0f (t1) dt1 −

∫ t−tp

0f (t1) dt1

](A3)

from which, we finally have:

Q(t) = p AT[S(t) − S

(t − tp

)]. (A4)

Q(t) is a continuos function oft at t = tp.Thus, the basin response Eq. (1) can be expressed as

Q(t) =

{p AT S(t)

(0 ≤ t ≤ tp

)p AT

(S(t) − S

(t − tp

)) (t > tp

),

(A5)

whereS(t) is the the S-hydrograph (Doodge, 2003). No-tice thatS(t) is a continuos function of time andQ(t) hasa possible discontinuous derivative att = tp. The maximumdischarge is obtained at the time to peak,t∗, which is foundby solving the equationdQ/dt = 0 (Henderson, 1963):

dS(t)

dt= f (t) = 0 0 ≤ t ≤ tp

dS(t)

dt=

dS(t − tp

)dt

t > tp. (A6)

Notice that the solution of Eq. (A6) might not correspond tothe actual maximum discharge. In fact, the maximum couldoccur in correspondence of the discontinuity point (i.e.,t∗ =

tp) in the first-order derivative ofQ(t) (see Equation (A5).Therefore, there is the need to check which one between thesolution of Eq. (A6) and t∗ = tp corresponds to an actualmaximum inQ(t).

By definition in Eq. (A6a) f (t) is null for t∗ = 0 – whichrepresents a trivial and unphysical solution – or fort∗ = τc.Thus, if tp > τc the time to peak coincides with the concen-tration time, otherwise (i.e. fortp ≤ τc) the solution is foundby solving Eq. (A6b), which is equivalent to Eq. (4).

Appendix B

An interesting application of Henderson’s equation is foundfor the case of the Nash IUH (Nash, 1957):

f (t) =1

n!

(t

k

)n−1

e−t/k (B1)

wheren andk are two calibration parameters. Forn = 1 thehydrograph is a negative exponential (linear reservoir) andthe peak is attp (i.e. t∗ = tp). Forn ≥ 2, Eq. (4) becomes

(1 −

tp

t

)n−1

= e−tp/k (B2)

which is solved as:

t∗ =tp

1 −(exp

(−tp/k

))1/(n−1). (B3)

It is easy to observe thatt∗ is always greater thantp and thatt∗ is an increasing function of the parametern.

The resulting critical rainfall time is determined (Eq.9) bysolving:

m =tp (t∗(n−1)) et∗/k

0(n,(t∗ − tp

)/k)

− 0(n, t∗/k)(B4)

where0 is the incomplete gamma function:

0(a,x) =

∫∞

0ta−1 e−t dt. (B5)

As noted, the linear-reservoir model (n = 1 case), is a par-ticular case of Eq. (B2). In this case,1t = 0 for any tp. Inthis case Eq. (9) becomes

m =

(tp/k

)e−tp/k

1 − e−tp/k. (B6)



Appendix C

The hydrograph response is expressed by Eq. (1) with p

given by Eq. (3) andf (t −τ) by Eq. (11):

Q(t) = AT p(tp, tr

) ∫ t

0H(tp − τ

)×

∫ L

0W(x) f (t − τ |x) dx d τ. (C1)

Whenf (t−τ |x) is expressed by Eq. (17), the change of vari-ablet −τ→t ′ leads to

Q(t)

p(tp, tr

)AT

=

∫ L

0 dx W(x)∫ t

0 f (t ′|x) dt ′

=∫ L

0 dx W(x)2(t |x)

for 0 ≤ t ≤ tp∫ L

0 dx W(x)∫ t

t−tpf (t ′|x) dt ′

=∫ L

0 dx W(x)[2(t |x)−2

(t − tp|x

)]for t > tp

(C2)

where

2(t) =

∫ t

0f (t ′|x) dt ′ = L−1

[f (s|x)

s

]t ′=t

. (C3)

In Eq. (C3) L−1[] represents the inverse Laplace-transform,

while f (s|x) is the Laplace transform of Eq. (17)

f (s|x) = Exp

[x

u −√

u2 + 4 s D

2 D

]. (C4)

Equation (C4) in Eq. (C3) gives

2(t) = Exp( u x

2 D

)L−1

Exp(−a

√s + b2

)s

(C5)

wherea = x/√

D and b = u/2√

D. The inversion of theLaplace transform leads to Eq. (19).

Acknowledgements.We thank the two anonymous reviewers forhaving contributed to the improvement of the original manuscriptwith their valuable comments. The research was partially supportedby the Italian Space Agency within the MORFEO project. Figureswere produced by using the set of DEM analysis tools developedby Rigon et al.(2006) and using the Open Source JGrass GISsoftware (http://www.jgrass.org). The model PeakFlow is freelyavailable for download under GPL 3.0 license as part of theGIS JGrass (where it is coded as a OpenMI 1.4 component) andin the JGrasstools (http://www.jgrasstools.org) coded as a OMS3component.

Edited by: F. Laio

References

Barling, R. D., Moore, I. D., and Grayson, R. B.: A quasi-dynamicwetness index for characterise the spatial distribution of zones ofsurface saturation and soil-water content, Water Resour. Res., 30,1029–1044, 1994.

Bell, F. C.: Generalized rainfall duration-frequency relationships, J.Hydraul. Eng., 95(HY1), 311–327, 1969.

Beven, K. J. and Kirkby, M. J.: A physically-based variable con-tributing area model of basin hydrology, Hydrol. Sci. Bull., 24,43–69, 1979.

Beven, K. J.: Rainfall-Runoff Modelling: the Primer, Wiley, Chich-ester, 360 pp., 2001.

Beven, K. J. and Wood, E. F.: Flow routing and the hydrologicalresponse of channel networks, in: Channel Network Hydrology,edited by: Beven, K. J. and Kirkby, M. J., Wiley, Chichester,Chap. 5., 1993.

Botter, G. and Rinaldo, A.: Scale effect on geomorphologicand kinematic dispersion, Water Resour. Res., 39(10), 1286,doi:10.1029/2003WR002154, 2003.

Brutsaert, W.: Hydrology: an introduction, Cambridge universityPress, Cambridge, UK, 2005.

Burlando, P. and Rosso, R.: Scaling and multiscaling Depth-Duration-Frequency curves of storm precipitation, J. Hydrol.,187(1–2), 45–64, 1996.

Chow, V. T., Maidment, D. R., Mays, L. W.: Applied Hydrology,McGraw-Hill, 1988.

Da Ross, D. and Borga, M.: Use of digital elevation model data forthe derivation of the geomorphological instantaneous unit hydro-graph, Hydrol. Process., 11, 13–33, 1997.

Doodge, J. C. I.: The rational method for estimating flood peaks,Engineering, 184, 311–313, 374–377, 1957.

Doodge, J. C. I.: Linear theory of hydrologic systems, EGU reprintsseries, Katlenburg-Lindau, Germany, 1, 2003.

D’Odorico, P. and Rigon, R.: Hillslope and channel contributions tothe hydrologic response, Water Resour. Res., 39(5), 1113–1121,doi:10.1029/2002WR001708, 2003.

D’Odorico, P., Fagherazzi, S., and Rigon, R.: Potential for land-sliding: dependence on hyetograph characteristics, J. Geophys.Res.-Earth, 110, F01007,doi:10.1029/2004JF000127, 2005.

Fiorentino, M., Rossi, F. , and Villani, P.: Effect of the basin geo-morphoclimatic characteristics on the mean annual flood reduc-tion curve, inProceedings of the IASTED International Confer-ence, Modeling and Simulation, pp. 17771784, Pittsburgh, Pa.,1987.

Franchini, M. and O’Connell, P. E.: An analysis of the dynamiccomponent of the geomorphologic instantaneous unit hydro-graph, J. Hydrol., 175(1–4), 407–428, 1996.

Gupta, V. J. and Mesa, O. J.: Runoff generation and the hydrologicresponse via channel network geomorphology – recent progressand open problems, J. Hydrol., 102, 3–28, 1988.

Gupta, V. J., Waymire, E., and Wang, C. T.: A representation ofan IUH from geomorphology, Water Resour. Res., 16, 885–892,1980.

Henderson, F. M.: Some Properties of the Unit Hydrograph, J. Geo-phys. Res., 68(16), 4785–4794, 1963.

Hershfield, D. M.: Rainfall frequency atlas of the United States fordurations from 30 minutes to 24 hours and return periods from1 to 100 years, Technical Paper No. 40, superseded by Hydrom-eteorological Report No. 35, US Department of Commerce, Na-


http://www.jgrass.org

http://www.jgrasstools.org

http://dx.doi.org/10.1029/2003WR002154

http://dx.doi.org/10.1029/2002WR001708

http://dx.doi.org/10.1029/2004JF000127


tional Weather Service, Silver Springs, MD, 1961.Kirkby, M.: A runoff simulation model based on hillslope topog-

raphy, in: Scale problem in Hydrology, edited by: Gupta, V. K.,Rodriguez-Iturbe, I., and Wood, E., and Reidel, D., Dordrecht,1986.

Iacobellis, V. and Fiorentino, M.: Derived distribution of floodsbased on the concept of partial area coverage with a climatic ap-peal, Wat. Resour. Res., 36, 469–482, 2000.

Mesa, O. J. and Mifflin, E. R.: On the relative role of hillslope andnetwork geometry in hydrologic response, in: Scale problem inHydrology, edited by: Gupta, V. K., Rodriguez-Iturbe, I., andWood, E., and Reidel, D., Dordrecht, 1986.

Mulvaney, T. J.: On the use of self-registering rain and flood gaugesin making observation of the relations of rainfall and flood dis-charges in a given catchment, Proceedings of the Institution ofCivil Engineers of Ireland, 4, 19–31, 1851.

Meynink, W. J. C. and Cordery, I.: Critical duration of rainfall forFlood estimation, Water Resour. Res., 12(6), 1209–1214, 1976.

Naden, P. S.: Spatial variability in flood estimation for large catch-ments: the exploitation of channel network structure, Hydrolog.Sci. J., 37, 53–71, 1992.

Naden, P., Broadhurst, P., Tauveron, N., and Walker, A.: River rout-ing at the continental scale: use of globally-available data andan a priori method of parameter estimation, Hydrol. Earth Syst.Sci., 3, 109–123,doi:10.5194/hess-3-109-1999, 1999.

Nash, J. E.: The form of the instantaneous unit hydrograph, Inter-national Association of Science and Hydrology, 45(3), 114–121,1957.

Rigon, R., Ghesla, E., Tiso, C. R., and Cozzini, A.: The Hor-ton machine: a system for DEM analysis: the reference man-ual, Dipartimento di Ingegneria Civile ed Ambientale and CU-DAM, http://www.ing.unitn.it/dica/eng/Quaderni/index.php, lastaccess: 28 April 2011, 1–144, 2006.

Rinaldo, A. and Rodriguez-Iturbe, I.: Geomorphological theoryof the hydrological response, Hydrol. Process., 10(6), 803–829,1996.

Rinaldo, A., Marani, A., and Rigon, R.: Geomorphological disper-sion, Water Resour. Res., 27(4), 513–525, 1991.

Rinaldo, A., Vogel, G. K., Rigon, R., and Rodriguez-Iturbe, I.:Can one gauge the shape of a basin?, Water Resour. Res., 31(4),1119–1127, 1995.

Robinson, J. S. and Sivapalan, M.: An investigation into the phys-ical causes of scaling and heterogeneity of regional flood fre-quency, Water Resour. Res., 33(5), 1045–1059, 1997.

Rodriguez-Iturbe, I. and Valdes, J. B.: The geomorphic structure ofhydrologic response, Water Resour. Res., 18(4), 877–886, 1979.

Ross, C. N.: The calculation of flood discharge by the use of timecontour plan isochrones, Transaction of the Institute of Engi-neers, Australia, 2, 85–95, 1921.

Saco, P. M. and Kumar, P.: Kinematic dispersion in stream net-works, Part 1: Coupling hydraulic and network geometry, Wa-ter Resour. Res., 38(11), 1244,doi:10.1029/2001WR000695,2002a.

Saco, P. M. and Kumar, P.: Kinematic dispersion in stream net-works, Part 2: Scale issues and self-similar network organization,Water Resour. Res., 38(11), 1245,doi:10.1029/2001WR000694,2002b.

Sherman, L. K.: Streamflow from rainfall by the unit-graph method,Engineering News-Record, 108, 501–505, 1932.

Shreve, R. L.: Stream lengths and basin areas in topologically ran-dom channel networks, J. Geol., 77, 397–414, 1969.

Sivapalan, M., Beven, K. J., and Wood, E. F.: On the hydrologicsimilarity, 2. A scaled model of storm runoff production, WaterResour. Res., 23(12), 2266–2278, 1987.

Snell, J. D. and Sivapalan, M.: On geomorphological dis-persion in natural catchments and the geomorphologicalunit hydrograph, Water Resour. Res., (30(7), 2311–2324,doi:10.1029/94WR00537, 1994.

Wood, E. F., Sivapalan, M., and Beven, K.: Similarity in smallcatchments storm response, Rev. Geophys., 28, 1–18, 1990.

Woods, R. A. and Sivapalan, M.: A connection between topograph-ically driven runoff generation and channel network structure,Water Resour. Res., 33(12), 2939–2950, 1997.

Woods, R. A. and Sivapalan, M.: A synthesis of space-time vari-ability in storm response: rainfall, runoff generation and routing,Water Resour. Res., 35(8), 2469–2485, 1999.

Yang, D., Herath, S., and Musiake, K.: A hillslope-based hydrolog-ical model using catchment area and width functions, Hydrolog.Sci. J., 1, 49–65, 2002.


http://dx.doi.org/10.5194/hess-3-109-1999

http://www.ing.unitn.it/dica/eng/Quaderni/index.php

http://dx.doi.org/10.1029/2001WR000695

http://dx.doi.org/10.1029/2001WR000694

http://dx.doi.org/10.1029/94WR00537

The geomorphic structure of the runoff peak

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